Test 5 Review

Quadratic Equations

100

How many real solutions does the following equation have?

x² - 49 = 0

Two real solutions!

b² - 4ac = 0² - 4(1)(-49) = 196 > 0

200

How many real solutions does the following equation have?

x² + 3x + 4 = 0

No real solutions!

b² - 4ac = 3² - 4(1)(4) = 9 - 16 = -7 < 0

300

Solve the following:

5x + 3x² = 8

x = -8/3, 1

400

Solve the following:

3x² - 6x - 12 = 0

x = 1 +√5, 1 - √5

500

Solve the following:

8x² + 4x - 16 = -x² 

x = (-2 ± 2√37)/9

Radical Expressions and Equations

100

Simplify: √(100v³)

10v√(v)

200

Simplify:

-3√18 + 3√8 - √24

-3(3√2) + 3(2√2) - 2√6

-9√2 + 6√2 - 2√6

-3√2 - 2√6

300

Simplify:

1 / (8 - 2√5)

(4 + √5) / 22

400

Solve:

√(x - 4) = 3

x - 4 = 9

x = 13

500

Solve: 

√(30 - x) = x

30 - x = x² 

0 = x² + x - 30

0 = (x - 5)(x + 6)

x = 5, -6  —> -6 doesn’t work!

Solution: x = 5

Rational Functions

100

What is the vertical asymptote for the following function?  Give the equation.

y = 2/x

x = 0

200

What is the horizontal asymptote for the following function? Give the equation.

y = 1/x

y = 0

300

What is the vertical asymptote for the following function? Give the equation.

y = (2x) / (x² - 9)

x = 3, x = -3

x² - 9 = 0

x²  = 9

x = ±3

400

What is the horizontal asymptote for the following function? Give the equation.

y = 2/(x+1) + 3

y = 3

500

What is the horizontal asymptote for the following function?  Give the equation.

(x³ - 1) / (x² - 9)

Degree of numerator > degree of denominator

There is no horizontal asymptote!

Piece-wise Defined Functions

100

Evaluate f(3).

f(x) = 

x - 1    if x ≤ 0

2x       if x > 0

f(3) = 2(3) = 6

200

Evaluate f(-1)

f(x):=

2x²   if x ≤ 0

3x     if x > 0

f(-1) = 2(-1)² = 2(1) = 2

300

Evaluate f(-121).

f(x):=

-x.   if x ≥ 0

x.    if x < 0

f(-121) = -121

400

Evaluate g(1).

g(x) =

x + 8.     if x ≤ 1

3 - x.      if x > 1

g(1) = 1 + 8 = 9

500

Evaluate f(4)

f(x)=

x²             if x ≤ -4

2x³  + 1    if -4 < x ≤ 4

3x - 2.      if x > 4

f(4) = 2(4)³ + 1 = 2(64) + 1 = 128 + 1 = 129

Grab Bag

100

What is the value of the discriminant for the following quadratic?

3x² + 5x - 18 = 0 

b² - 4ac = 5² -4(3)(-18) = 241

200

What is the value of the discriminant for the quadratic?

-6x² - 3x = -4  

b² - 4ac = (-3)² -4(-6)(4) = 9 - 96 = 105

300

Simplify:

2√32 + 3√8 - 4√18

2(4√2) + 3(2√2) - 4(3√2) = 2√2

400

Simplify:

√(2x³y)√(12xy)

√(24x⁴y²) = 2x²y√6 

500

Simplify:

(√2 - 5)² 

(√2 - 5)(√2 - 5) = 2 - 5√2 - 5√2 + 25 = 27 - 10√2